Integrand size = 45, antiderivative size = 25 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=2+\log (x)-\frac {1}{25} x^2 \left (6+x^2-\frac {3 \log (x)}{x}\right )^2 \]
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Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 14, 2404, 2332, 2338, 2341} \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=-\frac {x^6}{25}-\frac {12 x^4}{25}+\frac {6}{25} x^3 \log (x)-\frac {36 x^2}{25}-\frac {9 \log ^2(x)}{25}+\frac {36}{25} x \log (x)+\log (x) \]
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Rule 12
Rule 14
Rule 2332
Rule 2338
Rule 2341
Rule 2404
Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{x} \, dx \\ & = \frac {1}{25} \int \left (\frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6}{x}+\frac {18 \left (-1+2 x+x^3\right ) \log (x)}{x}\right ) \, dx \\ & = \frac {1}{25} \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6}{x} \, dx+\frac {18}{25} \int \frac {\left (-1+2 x+x^3\right ) \log (x)}{x} \, dx \\ & = \frac {1}{25} \int \left (36+\frac {25}{x}-72 x+6 x^2-48 x^3-6 x^5\right ) \, dx+\frac {18}{25} \int \left (2 \log (x)-\frac {\log (x)}{x}+x^2 \log (x)\right ) \, dx \\ & = \frac {36 x}{25}-\frac {36 x^2}{25}+\frac {2 x^3}{25}-\frac {12 x^4}{25}-\frac {x^6}{25}+\log (x)-\frac {18}{25} \int \frac {\log (x)}{x} \, dx+\frac {18}{25} \int x^2 \log (x) \, dx+\frac {36}{25} \int \log (x) \, dx \\ & = -\frac {36 x^2}{25}-\frac {12 x^4}{25}-\frac {x^6}{25}+\log (x)+\frac {36}{25} x \log (x)+\frac {6}{25} x^3 \log (x)-\frac {9 \log ^2(x)}{25} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=-\frac {36 x^2}{25}-\frac {12 x^4}{25}-\frac {x^6}{25}+\log (x)+\frac {36}{25} x \log (x)+\frac {6}{25} x^3 \log (x)-\frac {9 \log ^2(x)}{25} \]
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Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48
method | result | size |
default | \(-\frac {x^{6}}{25}+\frac {6 x^{3} \ln \left (x \right )}{25}-\frac {12 x^{4}}{25}+\frac {36 x \ln \left (x \right )}{25}-\frac {36 x^{2}}{25}-\frac {9 \ln \left (x \right )^{2}}{25}+\ln \left (x \right )\) | \(37\) |
norman | \(-\frac {x^{6}}{25}+\frac {6 x^{3} \ln \left (x \right )}{25}-\frac {12 x^{4}}{25}+\frac {36 x \ln \left (x \right )}{25}-\frac {36 x^{2}}{25}-\frac {9 \ln \left (x \right )^{2}}{25}+\ln \left (x \right )\) | \(37\) |
parallelrisch | \(-\frac {x^{6}}{25}+\frac {6 x^{3} \ln \left (x \right )}{25}-\frac {12 x^{4}}{25}+\frac {36 x \ln \left (x \right )}{25}-\frac {36 x^{2}}{25}-\frac {9 \ln \left (x \right )^{2}}{25}+\ln \left (x \right )\) | \(37\) |
parts | \(-\frac {x^{6}}{25}+\frac {6 x^{3} \ln \left (x \right )}{25}-\frac {12 x^{4}}{25}+\frac {36 x \ln \left (x \right )}{25}-\frac {36 x^{2}}{25}-\frac {9 \ln \left (x \right )^{2}}{25}+\ln \left (x \right )\) | \(37\) |
risch | \(-\frac {9 \ln \left (x \right )^{2}}{25}+\frac {\left (6 x^{3}+36 x \right ) \ln \left (x \right )}{25}-\frac {x^{6}}{25}-\frac {12 x^{4}}{25}-\frac {36 x^{2}}{25}+\ln \left (x \right )-\frac {32}{25}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=-\frac {1}{25} \, x^{6} - \frac {12}{25} \, x^{4} - \frac {36}{25} \, x^{2} + \frac {1}{25} \, {\left (6 \, x^{3} + 36 \, x + 25\right )} \log \left (x\right ) - \frac {9}{25} \, \log \left (x\right )^{2} \]
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Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=- \frac {x^{6}}{25} - \frac {12 x^{4}}{25} - \frac {36 x^{2}}{25} + \left (\frac {6 x^{3}}{25} + \frac {36 x}{25}\right ) \log {\left (x \right )} - \frac {9 \log {\left (x \right )}^{2}}{25} + \log {\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=-\frac {1}{25} \, x^{6} - \frac {12}{25} \, x^{4} + \frac {6}{25} \, x^{3} \log \left (x\right ) - \frac {36}{25} \, x^{2} + \frac {36}{25} \, x \log \left (x\right ) - \frac {9}{25} \, \log \left (x\right )^{2} + \log \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=-\frac {1}{25} \, x^{6} - \frac {12}{25} \, x^{4} - \frac {36}{25} \, x^{2} + \frac {6}{25} \, {\left (x^{3} + 6 \, x\right )} \log \left (x\right ) - \frac {9}{25} \, \log \left (x\right )^{2} + \log \left (x\right ) \]
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Time = 13.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {25+36 x-72 x^2+6 x^3-48 x^4-6 x^6+\left (-18+36 x+18 x^3\right ) \log (x)}{25 x} \, dx=-\frac {x^6}{25}-\frac {12\,x^4}{25}+\frac {6\,x^3\,\ln \left (x\right )}{25}-\frac {36\,x^2}{25}+\frac {36\,x\,\ln \left (x\right )}{25}-\frac {9\,{\ln \left (x\right )}^2}{25}+\ln \left (x\right ) \]
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